Results 1 to 7 of 7

Math Help - Differentiable function

  1. #1
    Junior Member
    Joined
    Mar 2010
    Posts
    45

    Differentiable function

    As h->0, show that(explain why)
    sin(x+h) = sinx +hcosx + o(h)
    Last edited by Monster32432421; April 28th 2010 at 05:41 PM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Apr 2005
    Posts
    16,419
    Thanks
    1855
    some thing like this always depends on what you already know and are allowed to use.

    If you know that the derivative of sin x is cos x, then you can find the tangent line to sin x at x= -x_0:

    sin(x)= cos(x_0)(x- x_0)+ sin(x_0)

    Taking x= x_0+ h, sin(x_0+ h)= cos(x_0)(h)+ sin(x_0) and, replacing x_0 with x,
    sin(x+ h)= cos(x)h+ sin(x).

    Another way would be to use the Taylor's series for sin(x) and cos(x):
    sin(x)= x- \frac{x^2}{2!}+ \frac{x^4}{4!}+ \cdot\cdot\cdot
    cos(x)= 1- \frac{x^3}{3!}+ \frac{x^5}{5!}+ \dot\cdot\cdot
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Mar 2010
    Posts
    45
    Could you explain this part? I'm not sure how you got the formula from just knowing the derivative of sin.. and how you got the tangent line at sinx at x= -x_0
    Quote Originally Posted by HallsofIvy View Post
    If you know that the derivative of sin x is cos x, then you can find the tangent line to sin x at x= -x_0:

    sin(x)= cos(x_0)(x- x_0)+ sin(x_0)
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    Apr 2005
    Posts
    16,419
    Thanks
    1855
    Quote Originally Posted by Monster32432421 View Post
    Could you explain this part? I'm not sure how you got the formula from just knowing the derivative of sin.. and how you got the tangent line at sinx at x= -x_0
    Normally, one of the first things people learn in Calculus is that the derivative is the slope of the tangent line and that, if y= f(x), the tangent line at (x_0, f(x_0)) is y= f'(x_0)(x- x_0)+ f(x_0). Check your notes.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Junior Member
    Joined
    Mar 2010
    Posts
    45
    Quote Originally Posted by HallsofIvy View Post
    Normally, one of the first things people learn in Calculus is that the derivative is the slope of the tangent line and that, if y= f(x), the tangent line at (x_0, f(x_0)) is y= f'(x_0)(x- x_0)+ f(x_0). Check your notes.
    Another question...How come you are missing the o(h) in the final solution?

    thanks so much for your help so far..
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Junior Member
    Joined
    Mar 2010
    Posts
    45
    Sorry person who is apparently not that awesome at maths.

    "The above equation is wrong. The function on the LHS is the sine
    function, whereas on the RHS you have a straight line. How can they
    equal?

    The tangent line is NOT equal to the curve. They only meet at ONE point:
    the touching point."

    Could someone else do my question?....
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Senior Member
    Joined
    Feb 2010
    Posts
    422
    Do you know that if \lim_{x\to a} f(x) = L then f(x) = L + o(1) where the o(1) is as x\to a? Write down the difference quotient and use this to turn it into the tangent line estimate f(x+h) = f(x) + f'(x) h +o(h), then set f(x) = sin(x).

    Also, be nice.

    edit: Or were you trying to prove that the derivative of sine is the cosine?
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Is the following function differentiable?
    Posted in the Calculus Forum
    Replies: 6
    Last Post: December 14th 2011, 11:38 AM
  2. Replies: 0
    Last Post: October 3rd 2010, 08:03 AM
  3. Differentiable function on Rn
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: May 14th 2010, 05:56 AM
  4. Differentiable function
    Posted in the Differential Geometry Forum
    Replies: 0
    Last Post: November 13th 2009, 06:47 AM
  5. differentiable function
    Posted in the Calculus Forum
    Replies: 4
    Last Post: January 15th 2008, 05:30 AM

Search Tags


/mathhelpforum @mathhelpforum